Deriviation of WKB approximation

[Repetit]

Hey!

In deriving the WKB approximation the wave function is written as

<br /> \psi \left( x \right) = exp\left[ i S\left( x \right) \right ]<br />

Now, in some of the deriviations I've seen, the function S(x) is expanded as a power series in \hbar as

<br /> S(x) = S_0(x) + \hbar S_1(x) + \frac{\hbar}{2} S_2(x) ...<br />

I don't really understand this. It's something like S_0 being the classical result and, the next term being a first order quantum correction and so on. But why do you choose to expand in powers of \hbar? Can somebody explain to me what this is all about?

Thanks in advance
René

 

[quantumdude]

That particular form for S(x) has the correspondence principle built right into it. If you take the limit as \hbar \rightarrow 0, you recover the classical result.

 

[jhicks]

That particular form for S(x) has the correspondence principle built right into it. If you take the limit as \hbar \rightarrow 0, you recover the classical result.

(This thread appeared on Google and I have the exact same question) I am extremely confused at your statement. \hbar is a constant, right? How on Earth can one construct a power series of a function S(x) by expanding it as a function of a constant? What does that even mean?

I have taken a few (more like 1.5 and some self study) classes in QM on the engineering side, but this is over my head. I've currently borrowed a few different QM textbooks and they all say the same opaque thing.

 

[lbrits]

You'll be seeing a lot more constants being treated like parameters in physics, so you'll have to get used to it.

Lets parameterize all the possible universes by different values of \hbar, and solve quantum mechanics in each of them. Then you'd get a family of solutions parameterized by \hbar. If you choose our universe, corresponding to our \hbar, then in principle you have the solution to QM in our universe, no?

There's a caveat, of course. You have to assume that the solutions to problems behave smoothly with \hbar, which is a reasonable assumption, but only comes from experience.

Anyway, if you stick around long enough you'll get to differentiate with respect to orbital angular momentum \ell and all sorts of goodness (Feynman-Hellman theorem)

 

[jhicks]

I think I understand a little better now, but I'll try to explain what is bugging me still. After reading around, I've come to the conclusion a power series with respect to constants is not so far-fetched: For example, any decimal number can be expressed as a power series in 10, or any other number really.

However, the textbook I am primarily using ( Bransden & Joachain Quantum Mechanics: Second Edition ) mentions the power series for S(x) "does not converge, but is an asymptotic series for the function S(x). As a result, the best approximation to S(x) is obtained by keeping a finite number of terms". I've been reading about asymptotic series, but their rationale/use isn't very clear to me still.

 

[jhicks]

I guess the closest I can come to what an asymptotic series means in words, it's "Adding more terms to the expansion won't make the relative error appreciably smaller". Is this accurate?